What question did this study set out to answer?

This research examines the distribution of Fibonacci numbers modulo 24 and their relationship to discriminants.

June 26, 2026Open Access

Fibonacci Numbers Modulo 24 and the Heegner Discriminant Classes: An Equidistribution Theorem and a Prime-Index Density

Key Points

This research examines the distribution of Fibonacci numbers modulo 24 and their relationship to discriminants.
Identified residues of the Fibonacci sequence modulo 24 for odd indices.
Calculated natural densities of specific residue classes among odd indices and prime indices.
Utilized elementary proofs without modular forms or reliance on the Riemann Hypothesis.
Odd indices of the Fibonacci sequence yield residue set {1,2,5,10,13,17} each with density 1/6.
Residue set {5,13,17} occurs with density 1/2 among odd indices.
For prime indices, the density of residues {5,13,17} is specifically 3/4.

Abstract

We determine the exact behaviour of the Fibonacci sequence F (n) modulo 24. For odd indices n, the residues F (n) mod 24 lie in 1, 2, 5, 10, 13, 17, each with natural density exactly 1/6; in particular 5, 13, 17 occur with density exactly 1/2 (no bias within the admissible set). The set 5, 13, 17 is precisely the residues mod 24 of the six odd class-number-one discriminants (−7, −11, −19, −43, −67, −163). Over prime indices the density of F (p) ∈ 5, 13, 17 mod 24 is exactly 3/4, by the Pisano period and Dirichlet's theorem. All proofs are elementary; no modular-form mechanism and no claim regarding the Riemann Hypothesis. This version supersedes v1 (the earlier "≈58%" and "all nine" figures are withdrawn; correct values: 1/2 and six of nine).

Read Full Paperexternally

Ask AI

Helpful

Bookmark

View Full Paper